Chaos

Question 1

Can chaos exist in one dimension?

Answer 1

Yes, it can exist in systems from 1 to n-dimensions.

Question 2

In the analogy of a ball rolling down a hill, the valley represents a _________

Answer 2

stable steady state

Question 3

How is chaos defined in terms of a lyapunov exponent

Answer 3

Greater than 0

Question 4

Describe a chaotic orbit.

Answer 4

Sensitive dependence on initial conditions, so the eventual separation of the orbits of nearby initial conditions as the system moves forward in time. Does not tend toward periodicity, has a Lyapunov number greater than 1.

Question 5

When are lyapunov numbers and exponents undefined?

Answer 5

When the derivative of an orbit containing point x is equal to zero.

Question 6

When is an orbit asymptotically periodic?

Answer 6

If it converges to a periodic orbit as n approaches infinity.

Question 7

What is a stable manifold?

Answer 7

The set of points whose forward orbits converge to a fixed or periodic point.

Question 8

What is a basin of attraction?

Answer 8

Set of points whose orbits converge to an attracting fixed point or periodic point. Aka a sink.